The paper is the first of a series of research investigations on robust digital filters which refer to filters that offer optimal performance under variations of filter parameters. We begin with quantitative characterization of performance robustness of a digital filter against parameter uncertainties. This is followed by several properties of the proposed robust performance measures and design formulations of robust FIR filters in L₂ (least-squares) and L_∞ (minimax) sense as nonsmooth convex problems. We present an accelerated subgradient algorithm for the design of L_∞-robust FIR filters with technical details involved in implementing the proposed algorithm. A numerical example is included for illustration of the proposed design method and performance evaluation in comparison with conventional minimax FIR filters.